6 Application to Stellar Flares

It is well known that stellar flares and coronae have many similarities to solar flares and corona (e.g.,
Haisch, 1989; Güdel, 2002). Not only light curves of stellar flare emissions (in radio, H, visible
continuum, and X-rays) but also quantitative nature of flares such as time scale, plasma temperature,
density, and magnetic field strength are all similar, though the distribution of temperature and total energy
of stellar flares is much broader (, total energy ) than those of solar flares
(, total energy ).

It is believed that stellar flares are produced by the same mechanism, magnetic reconnection, as solar
flares. However, why do some of stellar flares show very high temperature and extremely large total energy?
Recent observations of young stars by X-ray satellites ASCA and ROSAT have revealed that young
stars such as protostars and T-Tauri stars frequently produce superhot flares with temperature
of 108 K (Koyama et al., 1996; Tsuboi et al., 1998; Imanishi et al., 2001, see Feigelson and
Montmerle, 1999 for a review). Time variation of X-ray intensity is similar to that of solar flares,
while the total energy released by those stellar flares amounts to 1036 – 1037 erg, much larger
than those of solar flares. Can these protostellar flares be explained on the basis of magnetic
reconnection?

A hint was given in a paper by Feldman et al. (1995). They show that there is a universal correlation
between flare temperature (T) and emission measure (EM) not only for solar flares but also for some of
stellar flares. Shibata and Yokoyama (1999) extended this universal correlation between T and EM and
apply it for solar microflares, T-Tauri star flares, and protostellar flares (see Figure 46). It
is remarkable that the correlation holds in a very wide range, and
. Shibata and Yokoyama (1999) then found that this universal correlation
can be explained by the simple scaling law (see Figure 46),

which is derived from the following three equations:

where is the magnetic field strength and is coronal density, both of which are in a normal state
(no occurrence of a flare), is the length of a flaring loop. Equation (46) is basically the same as
Equation (38).1

Figure 46 shows the observed correlation between the emission measure of solar and stellar flares and
their temperatures. It also shows the theoretical relation between the emission measure and temperature
given by the Equation (43) is plotted as solid lines for three cases of = 15, 50, 150 G in the case of
. Figure 46 shows that the observed correlation line corresponds to the line
of constant magnetic field strength within 30 – 150 G, and indeed the coronal magnetic field
strength is estimated to be about 40 – 300 G for solar and stellar flares. Similarly, if we eliminate
the magnetic field strength () from the Equations (43) and (46), we can plot the relation
between the emission measure and temperature for constant loop length, which is also shown
in Figure 46 as dash-dotted lines. We can see that the length of a solar microflaring loop is
108 – 109 cm, and the length of a solar flaring loop is 109 – 1010 cm. These are fully consistent with
observations.

It is interesting to see that the length of a stellar flaring loop is 1010 – 1012 cm, much larger than the
length of a solar flaring loop. This is consistent with observations that average field strength at the surface
of young stars is very strong, which is of order kilo Gauss (e.g., Johns-Krull et al., 1999), indicating that the
size of a coronal loop with strong magnetic field () is much larger than that in the
Sun.

The size of a flaring loop in young stars is estimated to be comparable to or even larger than the solar
radius ( 7 × 1010 cm). It should be noted here that the range of radius for these young stars is from 1
to 4 solar radii (e.g., Johns-Krull et al., 1999).

Consequently, we found the reason why some of stellar flares, especially young star flares, show very high
temperature and extremely large total energy, which is because the size of these flares is much larger than
that of solar flares. If the length of a flaring loop is larger, the flare temperature increases in
proportion to even if the magnetic field is the same, because the conduction cooling
() become less efficient for a longer loop. The total energy is simply determined by the total
magnetic energy contained in the corona in a normal state, , which explains the
observations very well, although only a fraction of this energy is available as we mentioned before
(Equation (1)).

Why can such a large coronal loop with strong magnetic field exist? Why is the filling factor of strong
magnetic fields large (near unity) in young stars? One possibility is that the protostar is just born,
keeping primordial magnetic field whose origin is in interstellar medium. The other possibility is
that the strong magnetic field with large filling factor is created by the dynamo action. Since
young stars rotate rapidly (more than 30 km s–1 which is much faster than the solar rotation,
2 km s–1), the dynamo action would be stronger. It is also expected that there is an accretion disk
(planet-forming disk) around a young star, so that strong interaction would occur between the
central stellar core and the surrounding disk, which may lead to magnetic reconnection. This
interacting process has been treated by Hayashi et al. (1996), who performed 2.5-dimensional
MHD simulations for the interaction of an accretion disk and stellar magnetosphere (dipole
magnetic field). They have shown that vigous magnetic reconnection associated with mass
ejection occurs. The reconnection releases a huge amount of magnetic energy up to the order
1036 erg (about 104 times more energetic than solar flares) stored in a sheared loop with a size of
.